A (k; g, h)-graph is a k-regular graph of girth pair (g, h) where g is the girth of the graph, h is the length of a smallest cycle of different parity than g and g < h. A (k; g, h)-cage is a (k; g, h)-graph with the least possible number of vertices denoted by n(k; g, h). In this paper we give a lower bound on n(k; g, h) and as a consequence we establish that every (k; 6)-cage is bipartite if it is free of odd cycles of length at most 2k - 1. This is a contribution to the conjecture claiming that every (k; g)-cage with even girth g is bipartite. We also obtain upper bounds on the order of (k; g, h)-graphs with g = 6, 8, 12. From the proofs of these upper bounds we obtain a construction of an infinite family of small (k; g, h)-graphs. In particular, the (3; 6, h)-graphs obtained for h = 7, 9, 11 are minimal. (C) 2013 Elsevier B.V. All rights reserved.

Let View the MathML source denote the symmetric digraph of a graph G. A 3-arc is a 4-tuple (y,a,b,x) of vertices such that both (y,a,b) and (a,b,x) are paths of length 2 in G. The 3-arc graphX(G) of a given graph G is defined to have vertices the arcs of View the MathML source, and they are denoted as (uv). Two vertices (ay),(bx) are adjacent in X(G) if and only if (y,a,b,x) is a 3-arc of G. The purpose of this work is to study the edge-connectivity and restricted edge-connectivity of 3-arc graphs. We prove that the 3-arc graph X(G) of every connected graph G of minimum degree d(G)=3 has ¿(X(G))=(d(G)-1)2. Furthermore, if G is a 2-connected graph, then X(G) has restricted edge-connectivity ¿(2)(X(G))=2(d(G)-1)2-2. We also provide examples showing that all these bounds are sharp. Concerning the vertex-connectivity, we prove that ¿(X(G))=min{¿(G)(d(G)-1),(d(G)-1)2}. This result improves a previous one by [M. Knor, S. Zhou, Diameter and connectivity of 3-arc graphs, Discrete Math. 310 (2010) 37–42]. Finally, we obtain that X(G) is superconnected if G is maximally connected.

A kernel of a digraph is a set of vertices which is both independent and absorbant. Let D be a digraph such that every proper induced subdigraph has a kernel. If D has a kernel, then D is kernel perfect, otherwise D is critical kernel-imperfect (for short CKI-digraph). In this work we prove that if a CKI-digraph D is not 2-arc connected, then D - a is kernel perfect for any bridge a of D. If D has no kernel but for all vertex x, D - x has a kernel, then D is called kernel critical. We give conditions on a kernel critical digraph D so that for all x 2 V (D) the kernel of D-x has at least two vertices. Concerning asymmetric digraphs, we show that every vertex u of an asymmetric CKI-digraph D on n = 5 vertices satisfies d+(u) + d-(u) = n - 3 and d+(u), d-(u) = n - 5. As a consequence, we establish that there are exactly four asymmetric CKI-digraphs on n > 7 vertices. Furthermore, we study the maximum order of a subtournament contained in a not necessarily asymmetric CKI-digraph.

Continuing the study of connectivity, initiated in §4.1 of the Handbook, we survey here some (sufficient) conditions under which a graph or digraph has a given connectivity or edge-connectivity. First, we describe results concerning maximal (vertex- or edge-) connectivity. Next, we deal with conditions for having (usually lower) bounds for the connectivity parameters. Finally, some other general connectivity measures, such as one instance of the so-called “conditional connectivity,” are considered.
For unexplained terminology concerning connectivity, see §4.1.

Connectivity is one of the central concepts of graph theory, from both a theoret- ical and a practical point of view. Its theoretical implications are mainly based on the existence of nice max-min characterization results, such as Menger’s theorems. In these theorems, one condition which is clearly necessary also turns out to be sufficient. Moreover, these results are closely related to some other key theorems in graph theory: Ford and Fulkerson’s theorem about flows and Hall’s theorem on perfect matchings. With respect to the applications, the study of connectivity parameters of graphs and digraphs is of great interest in the design of reliable and fault-tolerant interconnection or communication networks.
Since graph connectivity has been so widely studied, we limit ourselves here to the presentation of some of the key results dealing with finite simple graphs and digraphs. For results about infinite graphs and connectivity algorithms the reader can consult, for instance, Aharoni and Diestel [AhDi94], Gibbons [Gi85], Halin [Ha00], Henzinger, Rao, and Gabow [HeRaGa00], Wigderson [Wi92]. For further details, we refer the reader to some of the good textbooks and surveys available on the subject: Berge [Be76], Bermond, Homobono, and Peyrat [BeHoPe89], Frank [Fr90, Fr94, Fr95], Gross and Yellen [GrYe06], Hellwig and Volkmann [HeVo08], Lov ´asz [Lo93], Mader [Ma79], Oellermann [Oe96], Tutte [Tu66].

Let r, m, 2 = r < m and g = 3 be three positive integers. A graph with a prescribed degree set r, m and girth g having the least possible number of vertices is called a bi-regular cage or an (r, m; g)-cage, and its order is denoted by n(r, m; g). In this paper we provide upper bounds on n(r, m; g) for some related values of r, m and even girth g at least 8. Moreover, if r - 1 is a prime power and m = 5, we construct the smallest currently known (r, m; 8)-graphs. Also, if r = 3 and m = 7 is not divisible by 3, we prove that n(3,m;8) = [25m/3] + 7. Finally, we construct a family of (3, m; 8)-graphs of order 9m + 3 which are cages for m = 4,5,7.

Under mild restrictions, we characterize all ways in which an incidence graph of a biaffine plane over a finite field can be extended to a vertex-transitive graph of diameter 2 and a given degree with a comparatively large number of vertices.

A (k;g)(k;g)-cage is a kk-regular graph of girth gg with minimum order. In this work, for all k=3k=3 and g=5g=5 odd, we present an upper bound of the order of a (k;g+1)(k;g+1)-cage in terms of the order of a (k;g)(k;g)-cage, improving a previous result by Sauer of 1967. We also show that every (k;11)(k;11)-cage with k=6k=6 contains a cycle of length 12, supporting a conjecture by Harary and Kovács of 1983.

Let 2 <= r < m and g be positive integers. An ({r, m}; g)-graph (or biregular graph) is a graph with degree set {r, m} and girth g, and an ({r, m}; g)-cage (or biregular cage) is an ({r, m}; g)-graph of minimum order n({r, m}; g). If m = r +1, an ({r,m};g)-cage is said to be a semiregular cage.
In this paper we generalize the reduction and graph amalgam operations from [M. Abreu, G. Araujo Pardo, C. Balbuena, D. Labbate. Families of Small Regular Graphs of Girth 5. Discrete Math. 312(18) (2012) 2832-2842] on the incidence graphs of an affine and a biaffine plane obtaining two new infinite families of biregular cages and two new semiregular cages. The constructed new families are ({r, 2r - 3}; 5)-cages for all r = q + 1 with q a prime power, and ({r, 2r - 5}; 5)-cages for all r = q + 1 with q a prime. The new semiregular cages are constructed for r = 5 and 6 with 31 and 43 vertices respectively.

The objective of this thesis is to study cages, constructions and properties of such families of graphs. For this, the study of graphs without short cycles plays a fundamental role in order to develop some knowledge on their structure, so we can later deal with the problems on cages. Cages were introduced by Tutte in 1947. In 1963, Erdös and Sachs proved that (k, g) -cages exist for any given values of k and g. Since then, large amount of research in cages has been devoted to their construction.
In this work we study structural properties such as the connectivity, diameter, and degree regularity of graphs without short cycles.
In some sense, connectivity is a measure of the reliability of a network. Two graphs with the same edge-connectivity, may be considered to have different reliabilities, as a more refined index than the edge-connectivity, edge-superconnectivity is proposed together with some other parameters called restricted connectivities.
By relaxing the conditions that are imposed for the graphs to be cages, we can achieve more refined connectivity properties on these families and also we have an approach to structural properties of the family of graphs with more restrictions (i.e., the cages).
Our aim, by studying such structural properties of cages is to get a deeper insight into their structure so we can attack the problem of their construction.
By way of example, we studied a condition on the diameter in relation to the girth pair of a graph, and as a corollary we obtained a result guaranteeing restricted connectivity of a special family of graphs arising from geometry, such as polarity graphs.
Also, we obtained a result proving the edge superconnectivity of semiregular cages. Based on these studies it was possible to develop the study of cages.
Therefore obtaining a relevant result with respect to the connectivity of cages, that is, cages are k/2-connected. And also arising from the previous work on girth pairs we obtained constructions for girth pair cages that proves a bound conjectured by Harary and Kovács, relating the order of girth pair cages with the one for cages. Concerning the degree and the diameter, there is the concept of a Moore graph, it was introduced by Hoffman and Singleton after Edward F. Moore, who posed the question of describing and classifying these graphs.
As well as having the maximum possible number of vertices for a given combination of degree and diameter, Moore graphs have the minimum possible number of vertices for a regular graph with given degree and girth. That is, any Moore graph is a cage. The formula for the number of vertices in a Moore graph can be generalized to allow a definition of Moore graphs with even girth (bipartite Moore graphs) as well as odd girth, and again these graphs are cages. Thus, Moore graphs give a lower bound for the order of cages, but they are known to exist only for very specific values of k, therefore it is interesting to study how far a cage is from this bound, this value is called the excess of a cage.
We studied the excess of graphs and give a contribution, in the sense of the work of Biggs and Ito, relating the bipartition of girth 6 cages with their orders. Entire families of cages can be obtained from finite geometries, for example, the graphs of incidence of projective planes of order q a prime power, are (q+1, 6)-cages. Also by using other incidence structures such as the generalized quadrangles or generalized hexagons, it can be obtained families of cages of girths 8 and 12.
In this thesis, we present a construction of an entire family of girth 7 cages that arises from some combinatorial properties of the incidence graphs of generalized quadrangles of order (q,q).